Next Article in Journal
Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics
Next Article in Special Issue
Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank
Previous Article in Journal / Special Issue
Norm Retrieval and Phase Retrieval by Projections
Article Menu

Export Article

Open AccessArticle
Axioms 2017, 6(2), 7; doi:10.3390/axioms6020007

Fourier Series for Singular Measures

Department of Mathematics, Butler University, Indianapolis, IN 46208, USA
Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA
Author to whom correspondence should be addressed.
Academic Editors: Palle E.T. Jorgensen and Azita Mayeli
Received: 21 February 2017 / Revised: 23 March 2017 / Accepted: 24 March 2017 / Published: 28 March 2017
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
View Full-Text   |   Download PDF [276 KB, uploaded 28 March 2017]


Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f L 2 ( μ ) possesses a Fourier series of the form f ( x ) = n = 0 c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = 0 1 f ( x ) e 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited. View Full-Text
Keywords: Fourier series; singular measure; Kaczmarz algorithm Fourier series; singular measure; Kaczmarz algorithm
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

MDPI and ACS Style

Herr, J.E.; Weber, E.S. Fourier Series for Singular Measures. Axioms 2017, 6, 7.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top